Non-PORC behaviour of a class of descendant $p$-groups

Abstract

We prove that the number of immediate descendants of order $p^{1}0$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture.